Inside logic #6

Modus Tollens

Suppose you are still at the airport and you are getting pretty bored. There is a television nearby. It looks like Britney Spears is on Larry King Live.

Larry: You look very nice today, Britney.

Britney: Thank you, Larry. Same to you.

Larry: So how about that quickie marriage you had?

B: Well that was a long time ago.

L: So are you married today or not?

B: Well, I got married again so now I am.

L: No kidding?

B: But I'm getting a divorce now.

L: Oh. It's hard to keep up.

B: I would prefer not to talk about any of that.

L: Ok. Well, ok then. Say, I have heard you own a lot of clothes. I was wondering about something.

B: What?

L: How do you go about deciding what to wear on any given day?

B: Well, its just whatever I feel like. Whatever colors and stuff feel right.

L: Ok.

B: Also, I use logic!

L: Logic? Really?

B: Sure. For example, this morning I thought about wearing my golden heels, the same ones I wore on the MTV awards show.

L: Hmm.

B: But then I thought. I am going to wear the golden blouse if I wear the golden heels. I always wear them together. But of course I wasn't going to wear that blouse again because I just wore it last week on David Letterman. Therefore, I knew I was not going to wear the golden heels!

L: So then you decided to wear ... what do you have on there?

B: These old red ones. I don't really like them, but I thought they'd be good enough for you.

L: Yes, they're fine.

B: Did you know that logic has forms?

L: Forms? What do you mean?

B: Well, when we reason logically, our thoughts have a certain form. For example, the way I just reasoned about the heels and the blouse? I reasoned the same way -- that means using the same form, that is, the same rule-- just last week after I did a pregnancy test.

L: A pregnancy test?!

B: Oops. Maybe I shouldn't have said that?

L: Well, sure. Its ok.

B: Ok.

L: But only if you want to talk about it.

B: I don't mind. On the side of the box it said something like that if you are pregnant, the indicator turns blue.

L: Is that right? What happened?

B: I had to urinate on it.

L: And?

B: And it didn't turn blue.

She's not pregnant!-- you think.

Unfortunately at that moment somebody changes the channel on the television over to a hockey game on ESPN.

--I wonder who? --

It really is none of your business. But at that moment a Breaking News Alert appears over the hockey game, BRITNEY NOT PREGNANT. BOYFRIEND UNKNOWN.

*

What should be pointed out here is that Britney's thinking is quite logical. Her reasoning about both the clothes and the pregnancy test is good --actually you and she collaborate in the reasoning about the test, since she offers the premises and you provide the conclusion after the channel got changed. The reasoning is valid because the conclusions, in each case, must be true if her premises are true. And she also is quite correct that the reasoning has the same form in the two cases.

Her reasoning in these two cases uses "if" and "not" together in a new way. The first premises in the reasoning about the clothes is:

I am going to wear the golden blouse if I wear the golden heels.

Notice that this is an "if/then" sentence even though the "if" part comes second, and it means the same thing as "If I wear the golden heels, then I am going to wear the golden blouse". The second premise is:

But I am not going to wear the golden blouse.

And her conclusion was

So I am not going to wear the golden heels.

It is intuitively obvious that this is a valid argument, because if the two premises are true, the conclusion must be true as well.

The first premise is a conditional that can be represented as H>B. The second premise is ~B, the negation of the consequent of H>B. Her conclusion is ~H, the negation of the antecedent. The reasoning about the pregnancy test has the same form. This form, of course, is not restricted to reasoning about clothes and pregnancy tests but represents a perfectly general form of good reasoning:

The rule MT, of course, can be used straightforwardly to prove the corresponding general sequent:

10 P>Q, ~Q } ~P

Combining the use of MT with DN we can prove the following new sequents.

11 P>~Q, Q } ~P

Notice that since the consequent of line 1, P>~Q, is ~Q, the rule MT can be applied only after we have obtained the negation of ~Q, which is ~~Q. And we do obtain it in line 3 by using the DN rule.

12 ~P>Q, ~Q } P